Topological Quantum Field Theories in Homotopy Theory...

Topological Quantum Field

Theories in Homotopy Theory

II

Ulrike Tillmann, Oxford

2016 V Congreso Latinoamericano de Matematicos

1

Leitmotif = Understanding Manifolds

1. Classical Cobordism Theory (Thom, ...)

2. Topological Field Theory (Witten, Atiyah, Segal, ...)

3. Cobordism Hypothesis (Baez-Dolan, Lurie, ...)

4. Classifying spaces of cobordism categories—— classifying space of cobordism categories. (Galatius–Madsen–Tillmann–Weiss)—— the cobordism hypothesis for invertible TQFTs—— filtration of the classical theory

5. Extracting information on Di↵(M)—— Mumford conjecture (Madsen–Weiss, ...)—— higher dimensional analogues. (Galatius–Randal-Williams, ...)

Manifolds

W is a manifold of dimension d if locally it is di↵eo-morphic to

Rd or Rd�0.

W is closed if it is compact and has no boundary.

Fundamental problem:– classify compact smooth manifolds W of dim d– understand their groups of di↵eomorphisms Di↵(W )– understand the topology of Mtop(W ) = BDi↵(W )– understand characteristic classes H⇤(Mtop(W ))

Topological moduli space

W closed of dimension d

Mtop(W ) := Emb(W,R1)/Di↵(W )

By Whitney, Emb(W,R1) ' ⇤ and hence

Mtop(W ) ' BDi↵(W )

• when W is oriented, the di↵eomorphisms preserve theorientation.• when @W 6= ;, the di↵eomorphisms fix a collar of theboundary.

Example 0: d = 0, W = n pts

Mtop(W ) = Emb(n pts, R1)/⌃n ' B⌃n

This is the configuration space of n unordered pointsin R1.

Example 1: d = 1, W = S1

Note that Di↵+(S1) ' S1 so that

Mtop(S1) ' BS1 ' CP1

Example 2: d = 2Compare with the moduli spaces of Riemann surfaces:

Mg = Tg/�g = R6g�6/�g

Note, for g > 1,

Di↵+(Fg)'�! ⇡0(Di↵+(Fg) =: �g

Hence, the forgetful map

Mtop(Fg) �!Mg

is a rational equivalence and

H⇤(Mtop(Fg);Q) = H⇤(Mg;Q)

Characteristic classes

There is a universal W -bundle on Mtop(W ), and anyfamily E of manifolds di↵eomorphic to W and parametrizedby a space X is given by a map

�E : X !Mtop(W )

Hence, the characteristic classes for W bundles aregiven by

H⇤(Mtop(W )) = H⇤(BDi↵(W ))

Construction of characteristic classes

Given a smooth orientable W -bundle

Wd ! E⇡! X

consider the vertical tangent bundle

Rd ! TvE :=a

x2XTEx ! E

It is classified by a map

�TvE : E ! BSO(d)

For every c 2 Hn(BSO(d)) consider

c(E) := ⇡!(�E(c)) 2 Hn�d(X)

Here ⇡! is the Gysin map. So

c 2 Hn�d(M(W )) = Hn�d(BDi↵(W ))

Example 2: d = 2

H⇤(BSO(2)) = H⇤(CP1) = Z[e]

For c = ei+1 2 H2i+2(CP1), c corresponds to the ithMorita-Mumford-Miller class

i := ei+1 2 H2i(Mg;Q)

Example 2: d = 2

H⇤(BSO(2)) = H⇤(CP1) = Z[e]

For c = ei+1 2 H2i+2(CP1), c corresponds to the ithMorita-Mumford-Miller class

i := ei+1 2 H2i(Mg;Q)

Homology stability [Harer, Ivanov, Boldsen, RW]For ⇤ < 2g/3, H⇤(B�g,n) is independent of g and n.

Mumford conjecture: For ⇤ (2g � 2)/3

H⇤(Mg;Q) ' Q[1,2, . . . ]

. early 1980s

4. Classifying space of cobordism categories

The enriched cobordism category Cobd:

Objects: R⇥`Md�1 M(M) ' `

Md�1 BDi↵(M)

Morphisms:

morCobd(M0,M1) '`Wd M(W ) ' `

Wd BDi↵(W ; @)

M0 �!W �M1

Example: d = 0

Objects: R as M�1 = ;

Morphisms:

morCobd(a0, a1) =a

nEmb(n pts, [a0, a1]⇥ R1)/⌃n

Classifying space functor

B : Topological Categories �! Spaces, C 7! BC

a a f b

a

f

b

g

c

gf

BC := (a

k�04k ⇥ {(f1, . . . , fk)|composable morphs})/ ⇠

For every object a in C, the adjoint of

41 ⇥morC(a, a) �! BCgives rise to a characteristic map

↵ : morC(a, a) �! maps([0,1], @;BC, a) = ⌦BC

Theorem [Galatius, Madsen, T., Weiss]

⌦B(Cob+d ) ' ⌦1MTSO(d) = limn!1⌦d+n((U?d,n)c)

where U?d,n is the orthogonal complement of the univer-

sal bundle Ud,n ! Gr+(d, n).

Note: Gr+(d, n) ,! Gr+(d, n+1) and U?d,n⇥R ,! U?d,n+1

Hence ⌃(U?d,n)c ! (U?d,n+1)c







Examples:

⌦B(Cob0) = ⌦1S1







Examples:

⌦B(Cob0) = ⌦1S1

⌦B(Cob+0 ) = ⌦1S1 ⇥⌦1S1

⌦B(Cob+1 ) = ⌦1+1S1

A similar theorem holds for any tangential structure ✓!

Excursion: tangential structures

Vectd(W ) = [W,BO(d)] = [W,Gr(d,1)]

E $ �E

Definition: Let ✓ : X ! BO(d) be a fiber bundle. A✓-structure on Wd is a lift of �TW : W ! BO(d) to X

Oriented +: Z/2Z! BSO(d)! BO(d)Framed fr: O(d)! EO(d)! BO(d)Background X: X ! BO(d)⇥X ! BO(d)Connected < k >: BO(d) < k >�! BO(d)

⌦1MT✓ = limn!1⌦d+n(✓⇤(U?d,n)c)

Example: ✓ : BO(d)⇥X ! BO(d)

⌦B(CobXd ) ' ⌦1MTSO(d)^X+ = limn!1⌦d+n((U?d,n)c^X+)

A ✓-structure on W is a map f : W ! X.

For d = 2, this gives a topological version ofGromov-Witten theory

Note: the cobordism category of manifolds in a back-ground space X gives rise to a generalised cohomology!

Lemma: H⇤(⌦10 MTSO(d),Q) ' ⇤⇤(H⇤>0(BSO(d);Q)[�d])


Proof: For fixed ⇤ > 0 and large n,

⇡⇤(⌦10 MTSO(d))⌦ Q = ⇡⇤( limn!1maps⇤(Sd+n, (U?d,n)c))⌦ Q

= ⇡⇤(maps⇤(Sd+n, (U?d,n)c)⌦ Q= ⇡⇤+d+n((Ud,n)

?)c ⌦ Q= H⇤+d+n((U

?d,n)

c)⌦ Q by Serre

= H⇤+d(Gr+(d, n))⌦ Q by Thom

= H⇤+d(BSO(d))⌦ QNow apply a theorem by Milnor-Moore on structure ofcommutative Hopf algebras and take duals.


For any closed d-dimensional manifold W

↵ : Mtop(W ) ⇢ morCobd(;, ;)! ⌦BCobd ' ⌦1MTSO(d)

For every c 2 H⇤+d(BSO(d))

↵⇤(c) = c 2 H⇤(Mtop(W );Q)

Example: For d = 2,

H⇤(BSO(2)) = H⇤(CP1) = Z[e], deg(e) = 2

So

H⇤(⌦10 MTSO(2),Q) = Q[1,2, . . . ], deg(i) = 2i

Fibration sequence

⌦1MTSO(d) !�! ⌦1⌃1(BSO(d)+) �! ⌦1MTSO(d�1).! is induced byU?d,n ! Ud,n ⇥Gr+(d,n) U

?d,n ' Gr+(d, n)⇥ Rd+n

Genauer proves that this corresponds to natural mapsof cobordism categories:

Cobd: d-dim cobordisms in [a0, a1]⇥ Rd+n�1 ⇥ (0,1)T

@ � Cobd: d-dim cobordisms in [a0, a1]⇥Rd+n�1⇥ [0,1)#Cobd�1: d� 1-dim cobordisms in [a0, a1]⇥Rd�1+n⇥ {0}

Application

Suppose Wd is an oriented manifold with boundary.Then restriction defines a map

r : BDi↵(W )! BDi↵(@W )

Let

c = pk11 . . . pkrr es 2 H⇤(BSO(d);Z) r = d/2 or (d+1)/2

Corollary [Giansiracusa-T.]

r⇤(c) = 0 when d even, or when d odd and s even.

In particular, 2i+1 goes to zero when restricted to thehandlebody subgroup Hg ⇢ �g.

Sketch of proofStarting with Madsen-Weiss, the proof has been con-tinuously simplified and the theorem generalised [Galatius–Madsen–T.–Weiss, Galatius, Galatius–Randal-Williams].

�d,n: space of all embedded d-manifolds without bound-ary which are closed as a subset in Rd+n; base point ?;topologized so that manifolds can disappear at infinity.

�kd,n: subspace of manifolds embedded in Rk⇥(0,1)n+d�k.

�d,n = �d+nd,n � · · · � �k

d,n � · · · � �0d,n '

a

W, @=?BDi↵(W )

Cobkd,n: k-fold cobordisms category of d-manifolds em-

bedded in Rd+n

limn!1 Cob1d,n = Cobd and limn!1 Cobdd,n = exCobd.

Step 1: (U?d,n)c ' �d+nd,n

tangential information: (P, v) 7! v � P .

Step 2: �kd,n

'�! ⌦�k+1d,n for k > 0

adjoint of R⇥�kd,n ! �k+1

d,n , (t,W ) 7!W+(0, . . . , t, . . . ,0)

Step 3: B(Cobkd,n) ' �kd,n

�1d,n ' B(Cob1d,n)

. �1d,n

' � Nq(P) '�! Nq(Cobd,n)

The fiber of the left arrow at W is the poset of regularvalues of the projection onto R; its nerve is contractible.

Step 1: (U?d,n)c ' �d+nd,n

tangential information: (P, v) 7! v � P .

Step 2: �kd,n

'�! ⌦�k+1d,n for k > 0

adjoint of R⇥�kd,n ! �k+1

d,n , (t,W ) 7!W+(0, . . . , t, . . . ,0)

Step 3: B(Cobkd,n) ' �kd,n

=) B(Cobkd,n) ' �kd,n ' · · · ' ⌦d+n�k�d+n

d,n ' ⌦d+n�k(U?d,n)c

for k = 1 and n!1, B(Cobd) ' ⌦1�1MTSO(d)for k = d+ n and n!1, B(exCobd) ' ⌦1�dMTSO(d)

An even finer filtration

⌦1MSO ' limn!1 lim

d!1⌦n(Un,d)

c

' limd!1

limn!1⌦n(U?d,n)c

' limd!1

limn!1B(Cobdd,n)

A 2-morphism in Cob22,1.

5. Unstable information

Question: Given Wd with ✓-structure, how close to anisomorphism in homology is ↵?

↵ : Mtop(W ) = BDi↵(W ) �! ⌦1MT✓(d)

5. Unstable information

Question: Given Wd with ✓-structure, how close to anisomorphism in homology is ↵?

↵ : Mtop(W ) = BDi↵(W ) �! ⌦10 MT✓(d)

Theorem [Barrett-Priddy, Quillen, Segal]

For d = 0: B⌃n↵�! ⌦1MTO(0) ' ⌦1S1 is a homol-

ogy isomorphism in degrees ⇤ n/2.


⌦B(Cob@d) ' ⌦B(Cobd) ' ⌦1MTSO(d)

Cob@d contains only cobordisms W with⇡0(W )! ⇡0(M1) surjective


⌦B(Cob@d) ' ⌦B(Cobd) ' ⌦1MTSO(d)

Cob@d contains only cobordisms W with⇡0(W )! ⇡0(M1) surjective

Previous result:

Theorem [T.] ⌦B(Cob@2) ' Z⇥B�+1

By homology stability, for ⇤ (2g � 2)/3

H⇤(B�+1) = H⇤(B�1) = H⇤(B�g)

Together these theorems imply

Theorem [Madsen-Weiss]

For d = 2: BDi↵(Fg)↵�! ⌦10 MTSO(2) is a homology

isomorphism in degrees ⇤ (2g � 2)/3.

=) Mumford’s Conjecture:

H⇤(Mg;Q)) ' H⇤(BDi↵(Fg),Q) ⇠ Q[1,2, . . . ]

Let Wg = ]gSd ⇥ Sd and d > 2. As Wg is (d � 1)-connected, it has a ✓ structure for

✓ : BO(2d) < d > �! BO(d)

Theorem [Galatius, Randal-Williams]For ⇤ < (g � 4)/2,

H⇤(BDi↵(Wg;D2d)) ' H⇤(BDi↵(Wg+1;D2d)

Theorem [Galatius, Randal-Williams]↵ : hocolimg!1BDi↵(Wg;D2d)! ⌦10 MT✓(d)is a homology equivalence

Similar results hold for more general simply connectedmanifolds of even dimension.

There are problems in odd dimensions!

Note

For d = 1: BDi↵(S1) ' CP1 ↵�! ⌦10 MTSO(1) '⌦1+1S1 is trivial in rational homology fro any W3.

There are problems in odd dimensions!

Note

For d = 1: BDi↵(S1) ' CP1 ↵�! ⌦10 MTSO(1) '⌦1+1S1 is trivial in rational homology fro any W3.

Theorem [Ebert]

For d = 3: BDi↵(W3) ↵�! ⌦1MTSO(3) is trivial inrational homology.

More generally, there are rational cohomology classesin the cohomology of ⌦1MTSO(2n + 1) that are notdetected by any 2n+1-dimensional manifold.

Qg = ]gSd ⇥ Sd+1

Theorem Perlmutter

For ⇤ (g � 3)/2,

H⇤(BDi↵(Qg;D2d+1) ' H⇤(BDi↵(Qg+1;D2d�1)

Hebestreit-Perlmutterconstruct a variant of the cobordism category that hasa good chance to play the role of Cobd in the odd di-mensional case.

Also true for other product of spheres.

General scheme of proof

1. Homology stability with respect to connected sumwith a suitable basic manifolds, e.g. Sd ⇥ Sd

2. Group completion argument to identify the ho-mology of the limit with the classifying space of asubcatgory of an appropriate cobordism category, e.g.limg!1H⇤BDi↵(Fg,D2) ' H⇤⌦BCob@2

3. Parametrised surgery to show that the subcate-gory has a classifying space that is homotopic to thewhole category, e.g. BCob@d ' BCobd

4. Identify the homotopy type of the classifying spaceof the whole category, e.g. ⌦BCobd ' ⌦1MTSO(d)

Multiplicative structure

⌃n ⇥⌃m �! ⌃n+m

Di↵(Fg,1)⇥Di↵(Fh,1) �! Di↵(Fg+h,1)

1 2 g

1 2 h

...

...

AutFn ⇥AutFm �! AutFn+m

. AutFn ' HtEq(VnS1)

Products

Gn ⇥Gm �! Gn+m

induce a monoid structure on

M :=G

n�0BGn

If products are commutative upto conjugation by anelement in Gn+m,

then the induced product on H⇤(M) is commutative

Group completion

Algebraic: M �! G(M) = Grothendieck group of MExample: N �! G(N) = Z

Homotopy theoretic:M �! ⌦BM = map⇤(S1, BM) = loop space of BM

• M = G a group =) ⌦BG ' G• M discrete =) ⌦BM ' G(M)

Group Completion Theorem:Let M =

Fn�0Mn be a topological monoid such that

the multiplication on H⇤(M) is commutative. Then

H⇤(⌦BM) = Z⇥ limn!1H⇤(Mn) = Z⇥H⇤(M1)

Wn H⇤-stab. ⌦B(tnM(Wn))

n pts n/2 ⌦1S1 Barratt-Eccles-Priddy

Sg,1 2g/3 ⌦1MTSO(2) Madsen-Weiss

Ng,1 g/3 ⌦1MTO(2) Madsen-Weiss & Wahl

VnS1 n/2 ⌦1S1 Galatius

#g(Sd ⇥ Sd)1 (g-4)/2 ⌦1MTSO(2d)hdi Galatius-Randal-Williams

(Sd ⇥Dd+1)1 (g-4)/2 Q(BO(2d+1)hdi) Botvinnik-Perlmutter

discrete (g-4)/2 ⌦1X�� Nariman

(Sd ⇥ Sd+1)1 (g-3)/2 candidate Perlmutter

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